where \(m=u-u_{xx}\) , \(u=u(t,x)\) , \(\rho =\rho (t,x)\) ; \(t\)  and \(x\)  denote time and space, respectively, whereas \(u\)  is the horizontal velocity of the fluid and \(\rho \)  is related to the horizontal deviation of the fluid's surface. This system was firstly reported by Olver and Rosenau, see [1]}, and later derived from the Green-Naghdi equations [2]} as a shallow water model by Constantin and Ivanov [3]}.
